The proof of (REF ) used the CP-chains method introduced by Furstenberg [1]}, and the local entropy averages and \(\rho \) -tree methods introduced by Hochman and Shmerkin [2]}. Such a mechanism turns out to be very useful to derive lower bounds of Hausdorff dimension of dynamically driven fractal measures. In particular it led Shmerkin [3]} and Wu [4]} to solve, independently, another conjecture of Furstenberg on intersections of invariant sets.
